"We just have to make sure we continue to work hard in the contest," he said. "We think we are in a very strong position to claim the nomination, but you have to give her credit, as we do."
Obama joked that he may consider a visit to "Saturday Night Live" to aid his campaign. Clinton received a ton of media coverage after she lampooned herself on the comedy show this weekend.
"I had fun the last time I was on 'Saturday Night Live.' If the opportunity comes up we'll have to talk," Obama said. "Maybe I'll even go on 'Saturday Night Live' when I become president."
Obama said his wife, Michelle, left the campaign and headed home to spend time with the couple's two young daughters.
"Being away from them is the hardest part of this campaign," Obama said. "They went to bed long before any of the results were out."
Obama added that his daughters were more concerned about receiving the puppy he promised them than whether he wins the nomination.
The former first lady fought hard for Latino voters and the white, working class voters who have typically supported her in earlier primaries.
The headline battles in delegate-rich Ohio and Texas were considered crucial primary contests that could cement Obama's hold on the Democratic presidential nomination, or help Clinton continue to fight on for her party's nomination.
In her victory speech in Columbus, Ohio, Tuesday night, Clinton claimed a comeback.
"For everyone here in Ohio and across America, who's ever been counted out but refused to be knocked out and for everyone who has stumbled but stood right back up, and for everyone who works hard and never gives up, this one is for you," Clinton told a cheering crowd of supporters.
"This nation's coming back and so is this campaign!" Clinton said.
In a subdued speech to supporters Tuesday night in San Antonio, Obama maintained that he is still ahead in the delegate count and slammed his Democratic rival.
"No matter what happens tonight, we have nearly the same delegate lead as we did this morning, and we are on our way to winning this nomination," Obama said.
"This is not the future we want. We want a new course for this country. We want new leadership in Washington. We want change in America," Obama told supporters. "John McCain and Sen. Clinton echo each other in dismissing this call for change."
On the Republican side, McCain swept Tuesday's GOP primaries in Ohio, Texas, Rhode Island and Vermont and now has the 1,191 delegates needed to become the Republican presidential nominee, according to ABC News' delegate count.
"I am very, very grateful [and] pleased to note my friends that tonight we have won enough delegates to claim with confidence, humility and a sense of great responsibility that I will be the Republican nominee for president of the United States," McCain told supporters Tuesday night.
"The contest begins tonight," he said to cheers and applause.
At McCain's election night party headquarters in Dallas, a huge banner reading "1191" hangs alongside hundreds of red, white and blue balloons.
McCain will meet today with President Bush, who is expected to endorse the Arizona senator at an event in the Rose Garden at the White House.
Conceding the race, former Gov. Mike Huckabee, R-Ark., suggested his candidacy gave a voice to the millions of Christian evangelical voters who supported his campaign.